Problem: Given $ m \angle ABC = 7x - 5$, $ m \angle CBD = 8x + 40$, and $ m \angle ABD = 110$, find $m\angle ABC$. $B$ $A$ $D$ $C$
Answer: From the diagram, we see that together ${\angle ABC}$ and ${\angle CBD}$ form ${\angle ABD}$ , so $ {m\angle ABC} + {m\angle CBD} = {m\angle ABD}$ Substitute in the expressions that were given for each measure: $ {7x - 5} + {8x + 40} = {110}$ Combine like terms: $ 15x + 35 = 110$ Subtract $35$ from both sides: $ 15x = 75$ Divide both sides by $15$ to find $x$ $ x = 5$ Substitute $5$ for $x$ in the expression that was given for $m\angle ABC$ $ m\angle ABC = 7({5}) - 5$ Simplify: $ {m\angle ABC = 35 - 5}$ So ${m\angle ABC = 30}$.